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Structural phase stability and electron-phonon coupling in lithium
Department of Physics, Georgetown University, Washington, DC 20057-0995 Lawrence Livermore National Laboratory, Livermore, California 94550 Department of Physics, Georgetown University, Washington, DC 20057-0995 Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 Department of Physics, Georgetown University, Washington, DC 20057-0995 First-principles calculations of the free energy of several structural phases of Li are presented. The density- functional linear-response approach is used to calculate the volume-dependent phonon frequencies needed forcomputing the vibrational free energy within the quasiharmonic approximation. We show that the transforma-tion from a close-packed structure at low temperatures to the bcc phase upon heating is driven by the largevibrational entropy associated with low-energy phonon modes in bcc Li. In addition, we find that the strengthof the electron-phonon interaction in Li is strongly dependent on crystal structure. The coupling strength issignificantly reduced in the low-temperature close-packed phases as compared to the bcc phase, and is con-sistent with the observed lack of a superconducting transition in Li. ͓S0163-1829͑99͒11905-2͔ I. INTRODUCTION
Neutron experiments3,8,9 show that while there may be The pressure-temperature structural phase diagram of some softening of certain phonon modes in bcc Li upon solid Li, arguably the simplest metal, is not well established.
cooling, the transformation occurs well before any phonon Even along the zero-pressure axis, Li exhibits some complex frequencies or elastic constants approach zero, ruling out the behavior. At ambient pressure and room temperature, Li soft-mode mechanism10–12 for the transformation. The trans- crystallizes in the bcc structure, but upon cooling, it under- formation more likely results from a competition between goes a martensitic transformation around 80 K. The transfor- the internal energies and entropies of the different phases.
mation was first observed in 1956,1 but the crystal structure For example, Friedel13 argued on general grounds that the of the low-temperature phase remained a subject of debate phonon spectrum should scale roughly with coordination for several decades. In 1984, Overhauser suggested that the number, leading to an overall lowering of the bcc phonon neutron scattering data were consistent with the 9R spectrum compared to that of close-packed structures. The structure,2 a close-packed phase with a nine-layer stacking resulting larger vibrational entropy associated with the bcc sequence. Subsequent investigations confirmed 9R as the pri- structure provides a qualitative explanation for the preva- mary structure at low temperatures,3 but also showed that it lence of the bcc structure as the high-temperature phase in is accompanied by a large and variable amount of a disor- metals. While these ideas are believed to apply to many sys- dered polytype consisting of short-ranged fcc- and hcp-type tems that exhibit martensitic transformations, it is useful to stacking order, as well as some amount of bcc.4 Furthermore, examine them in the context of detailed materials-specific upon heating, the 9R phase and disordered polytype appear phonon spectra which are accessible from first-principles cal- to transform first to an ordered fcc phase before reverting to Several first-principles theoretical investigations have fo- Little information is available about the phase diagram of cused on the relative stability of different crystal structures Li at higher pressures. Upon compression, the temperature at for Li at zero pressure.14,15 Conflicting results have emerged, which the martensitic transformation begins, M s , rises at a but all the studies agree that the energy differences between rate of about 20 K/GPa, at least up to pressures of 3 GPa,5,6 structures are small ͑much less than 1 mRy/atom͒, making it with the structural characteristics of the low-temperature difficult to determine the ordering reliably. Most of the the- phase remaining the same as at ambient pressure. At room oretical work has been based on zero-temperature calcula- temperature, a structural transition is observed at P tions that do not even include zero-point energies. Given that ϭ6.9 GPa.7 The crystal structure of the compressed phase Li is such a light atom, finite-temperature effects arising was initially identified as fcc, but the limited diffraction data from vibrational degrees of freedom are likely to play a role STRUCTURAL PHASE STABILITY AND ELECTRON- . . .
in the structural energetics of this system. In a recent study, cal density approximation ͑LDA͒. The Li pseudopotential is vibrational contributions to the free energies of bcc and fcc generated by the Troullier-Martins method.28 The Wigner Li were computed within the harmonic approximation.15 The interpolation formula is used for the correlation potential,29 fcc phase was found to be favored at Tϭ0 K, which is and a partial core correction is used to treat the nonlinearity consistent with experiments. Furthermore, the free-energy of the exchange and correlation interaction between the core difference between the two phases was calculated to decrease and valence charge densities.30 Since the structures lie very with increasing temperature, but, in contrast to what is ob- close in energy, large sets of k points are needed to reliably
served in experiments, the fcc phase was calculated to re- compare the total energies of different structures. We use main thermodynamically stable up to at least a few hundred 1120, 1122, 1300, and 918 points in the bcc, 9R, fcc, and hcp irreducible Brillouin zones ͑IBZ͒, respectively, to obtain to- In this paper, we go beyond the purely harmonic approxi- mation and include anharmonic effects at the quasiharmonic 10Ϫ5 Ry/atom with respect to BZ sampling. In addition, a level.16 This approximation, in which the volume depen- large plane-wave cutoff of 20 Ry is used in the calculation of dence of the phonon spectrum is explicitly included, hasbeen shown to provide a remarkably good description of the temperature dependence of the lattice constant and bulk The free energy at finite temperature has contributions modulus of bcc Li up to its melting point.17 The density- from both electronic excitations as well as lattice vibrations.
functional linear response method18,19 is used to compute the While the effects of thermal excitation of electrons are ex- volume-dependent phonon frequencies. We find that a quasi- pected to be weak at the temperatures of interest, their con- harmonic treatment of the phonon free energies reproduces tributions to the free energy can easily be included by using the experimentally observed bcc-9R transformation, though the transition temperature is overestimated. The thermody- standard DFT total energy.31 The lattice excitations are namic stabilization of the bcc structure at high temperatures treated within the quasiharmonic approximation,16 where the is shown to be completely attributable to its excess phonon full Hamiltonian at a given volume is approximated by a harmonic expansion about the equilibrium atomic positions, A possibly related issue is the lack of a superconducting but anharmonic effects are included through the explicit vol- transition in Li. Specific heat measurements,20 as well as ume dependence of the vibrational frequencies. The Helm- first-principles calculations21–25 suggest that the electron- phonon mass enhancement parameter in Li is ␭Ϸ0.45. Usingthe McMillan equation,26 with a Coulomb repulsion param- ͫ ͩប␻q␯͑V͒ͪͬ
eter of ␮*ϭ0.12, which is believed to be typical for simple F͑V,T͒ϭF ͑ el V , T ͒ ϩ k BT ͚ ln 2 sinh metals, we would expect to find a superconducting transition in Li at temperatures on the order of 1 K. Yet no transition where ␻q␯(V) is the frequency of the ␯th phonon band at the
has been observed in experiments, at least down to 6 mK.27 point q in the Brillouin zone.
Because of uncertainties in the low-temperature crystal The density-functional linear-response method is used to structure, most of the previous calculations, as well as the compute the phonon spectra for each structure at several dif- analysis of the specific heat data, were based on bcc Li. One ferent volumes. The dynamical matrices are computed using calculation of the electron-phonon coupling strength in 9R Li the scheme described in Ref. 19, where the self-consistent suggested that the coupling is indeed weaker in the low- change in the Hamiltonian caused by ionic displacements is temperature phase.24 However, there were large uncertainties obtained by solving a Bethe-Saltpeter equation for the in that result since it was based on frozen-phonon calcula- change in the charge density. At each volume, the dynamical tions in which it was only feasible to examine the coupling matrix is computed for 285, 240, 98, and 95 phonon wave constants at a few phonon wave vectors. Using the linear- vectors in the bcc, fcc, 9R, and hcp IBZs, respectively. A response approach rather than the frozen-phonon method, we planewave cutoff of 12 Ry is used in the calculation of pho- are able to efficiently sample wave vectors throughout theBrillouin zone, and more accurately calculate the structure dependence of the electron-phonon interaction in Li in thiswork. Our results show that in 9R, fcc, and hcp Li, ␭ is B. Results and discussion
significantly smaller than in bcc Li. We argue that the calcu- The equilibrium volumes and energies calculated for bcc, lated coupling in the close-packed phases is weak enough to fcc, hcp, and 9R Li are listed in Table I. For volumes within be consistent with current experimental limits on the transi- about Ϯ12% of the equilibrium volume, we find that the stacking of the close-packed planes in 9R Li is near ideal: The remainder of this paper is organized as follows. In when structural parameters are chosen to correspond to ideal Sec. II, we outline the methods used to compute the freeenergies and present our results for the structural phase sta- stacking, the calculated forces on the atoms are less than bility of Li. Section III focuses on the comparison of the 10Ϫ4 Ry/au and the differences between diagonal compo- electron-phonon interaction in bcc and 9R Li. A summary nents of the stress tensor are less than 2ϫ10Ϫ6 Ry/au3.
and concluding remarks are given in Sec. IV.
This is consistent with neutron data which show the c/a ratioto be within 0.1% of the ideal value.3 Similarly, for the hcp II. STRUCTURAL PHASE STABILITY OF LITHIUM
structure, the optimal value for c/a is found to be 1.63,which is very close to the ideal close-packed value of 1.633.
A. Method
The Tϭ0 K static calculations, which include only the The zero-temperature structural energetics are calculated LDA total energies, show bcc Li to be the least favorable of using the plane-wave pseudopotential method within the lo- all the structures considered. Bcc Li lies about 0.1 mRy/atom TABLE I. Equilibrium volume and energy calculated for bcc, fcc, 9R, and hcp Li. Results are presented for both static zero-temperature LDA calculations, as well as zero-temperature calcula-tions that include zero-point energies. For hcp Li, only static resultsare presented because full quasiharmonic calculations were not per-formed for this phase.
above the close-packed structures, all three of which havenearly the same energy. We find the lowest lying structure tobe fcc, with 9R and hcp Li only about 0.01 mRy/atom higher FIG. 1. Calculated Helmholtz free energy versus volume for bcc and 9R Li at two temperatures. The curves are fits of the calculatedpoints to the Birch equation of state.
The relative ordering of various structures for Li has been the subject of some controversy. Several static LDA studieshave been carried out, with varying results.14,15 Because the served experimentally. Interestingly, this close-packed to bcc different calculations differ in the basis sets employed, the transition was not found in a recent LDA calculation based sampling of the BZ, the form of the LDA potential, and the on the purely harmonic approximation.15 In that work, free shape approximations used, it is difficult to sort out the rea- energies for bcc and fcc Li were compared, and fcc Li was sons for the discrepancies. Nevertheless, all the calculations found to be energetically favored all the way from Tϭ0 K agree that the structures are close in energy, and hence are at up to several hundred kelvin. In the present work, we find least qualitatively consistent with the coexistence of various that the temperature range over which the close-packed phases at low temperatures. Given the structural similarity phase is favored indeed expands if the volume dependence of between the various close-packed phases, their near degen- the phonon spectrum is neglected, but we still find the bcc eracy in this monovalent s p metal is not surprising. While phase to be favored over both fcc and 9R Li at 300 K.
the calculated total-energy differences between the close- To pin down the transition temperature, it is useful to packed phases are too small to allow us to definitively deter- examine the Gibbs free energy at zero pressure. The differ- mine their relative ordering, we are confident in our result ence between the 9R and bcc zero-pressure Gibbs energies, that the close-packed phases are clustered close in energy, ⌬Gtot(Pϭ0,T), is plotted as a function of temperature in and that all are more stable than the bcc phase.
Fig. 2. The calculation indicates that 9R Li is thermodynami- The average phonon frequency is calculated to be about cally stable below about Tϭ200 K. Based on the uncertain- 1% lower in bcc Li than in the close-packed phases. Thus ties in the calculated energy differences, we estimate error when zero-point energies are included, the energy difference bars of about Ϯ50 K on the calculated transition tempera- between bcc Li and the close-packed phases decreases. Wefind however that the relative ordering of the structures re-mains the same. As can be seen in Table I, the inclusion ofzero-point energy also leads to an expansion in the equilib-rium volume. For bcc Li, the static LDA energies underesti-mate the volume by nearly 7%. Since the atoms are so light,vibrational effects are significant, and when zero point mo-tion is taken into account, the error in the calculated volumebecomes about 4.5%, which is more typical of the size oferrors found in LDA studies of other simple metals.
Since the close-packed phases are virtually indistinguish- able energetically, we choose the 9R structure, which is thepredominant low-temperature phase, as representative of allthe close-packed structures. For the remainder of this section,we focus on comparing the energetics of bcc and 9R Li.
The Helmholtz free energy for bcc and 9R Li are plotted as a function of volume in Fig. 1. Results are presented forTϭ0 K and Tϭ300 K. For each crystal structure, calcula- FIG. 2. Calculated difference in Gibbs free energy between 9R tions are carried out for six to eight volumes, and the results and bcc Li at zero pressure, plotted as a function of temperature.
are fit to the Birch equation of state.32 With increasing tem- The triangles show the difference in electronic contributions to the perature, the ordering of the phases reverses, and the bcc free energy and the squares show the difference in phonon contri- phase is calculated to be stable at room temperature, as ob- STRUCTURAL PHASE STABILITY AND ELECTRON- . . .
TABLE II. Electronic and electron-phonon-related properties computed for bcc, fcc, 9R, and hcp Li: the electronic density ofstates at the Fermi level N(EF), a logarithmic average of the pho-non frequencies ␻log , the electron-phonon mass enhancement pa-rameter ␭, and the minimum value of the Coulomb pseudopoten-tial, ␮* , that is consistent with the observed lack of a superconducting transition above 6 mK.
ture. Since the transformation actually takes place via nucle-ation and growth of the close-packed phase within the bccmatrix, the experimentally observed starting temperature forthe transition upon cooling, about 80 K, provides a lower FIG. 3. ͑a͒ Phonon density of states ͑per atom͒ and ͑b͒ electron- bound on the thermodynamic transition temperature. Simi- phonon spectral function calculated for bcc and 9R Li.
larly, the experimental starting temperature for the close-packed to bcc transformation upon heating, about 150 K, pletely by the excess vibrational entropy arising from the low-energy shear modes in the bcc phase.
The triangles in Fig. 2 show the 9R-bcc difference in the The Clausius-Clapeyron equation allows us to estimate electronic contributions to the free energy, ⌬G the slope of the phase line from the changes in entropy and volume at the transition: dT/d Pϭ⌬V/⌬Sϭ23 K/GPa at T⌬Sel . This quantity shows negligible temperature dependence and favors the close-packed 9R phase through- Pϭ0 GPa. This is in good agreement with the experimen- out the range of temperatures considered. In fact, ⌬G tally determined values for the pressure dependence of the dominated by the difference in internal energies, ⌬E onset temperature, which range from d M s /dTϭ15 K/GPa which is over an order of magnitude larger than T⌬S to 30 K/GPa.5,6 In principle, the entire phase line can be ϭ300 K. At low temperatures, the leading term in the elec- determined by analyzing the free energies at different pres- tronic entropy is proportional to the electronic density of sures. However, as noted in Refs. 34 and 15, the LDA pre- states at the Fermi level N(E dicts a negative Gruneisen parameter for the T1 monovalent metal, the electronic density of states depends in bcc Li, and an elastic instability at volumes corresponding weakly on crystal structure ͑Table II͒, and hence the elec- to pressures of PϾ2.5 GPa at Tϭ0 K ͑or PϾ3.1 GPa at tronic entropy plays a negligible role in the structural transi- Tϭ300 K). Experimentally, bcc Li indeed becomes un- stable upon compression, and transforms to a close-packed The squares in Fig. 2 show the difference between the 9R phase, but the transformation occurs at a higher pressure of and bcc phonon free energies. The difference in vibrational about 7 GPa. The underestimation of the transition pressure energies between bcc and 9R Li is virtually temperature in- indicates that anharmonic effects beyond the quasiharmonic dependent, at least up to Tϭ400 K, and it is the entropic level become increasingly important with compression, and a term in the free energy that stabilizes the bcc structure with treatment that includes phonon-phonon interactions that can increasing temperature. The phonon densities of states, F(␻), for the two phases are plotted in Fig. 3͑a͒. While themaximum phonon frequency is nearly the same in the two III. ELECTRON-PHONON COUPLING STRENGTH
structures, the bcc phonon spectrum has more weight in the IN LITHIUM
low-frequency regime ͑below about 15 meV͒. This enhanced A. Method
weight arises primarily from the T ͓␰␰ ¯ 0͔ polarization, and surrounding re- The matrix element for scattering of an electron from gions in the BZ. The entire T ͓␰␰ state nk to state nЈkЈ by a phonon with frequency ␻q␯ and
meV, and none of the close-packed structures have compa- eigenvector ⑀ˆq␯ is given by
rable low-energy phonon branches. These low-energy shearmodes in bcc Li are easy to excite and give large contribu- tions to the vibrational entropy. Over the range of tempera- g͑nk,nЈkЈ,q␯͒ϭͱ ប ͗nk͉⑀ˆ
nЈkЈ͘, ͑2͒
q␯• ␦ V SCF
tures considered, the vibrational entropy arising from phononmodes with frequency smaller than 8.5 meV is about twice where ␦VSCF is the self-consistent change in the potential as large in bcc Li than in 9R Li, while the entropy associated due to atomic displacements. This scattering gives rise to a with higher-frequency modes is about the same in the two structures. From the thermodynamic standpoint, the close-packed to bcc transformation in Li is therefore driven com- q␯ϭ 2 ␲␻ q␯͓ N͑ E F
gq␯͉2͘͘,
where the double brackets ͗͗ ͘͘ denote a doubly constrainedFermi surface average as defined in Ref. 36. This scatteringprocess also contributes to the effective mass of the electronsvia the mass enhancement parameter The total mass enhancement parameter, ␭, is obtained bysumming over branches and averaging over wave vectorsthroughout the Brillouin zone.
Within the Eliashberg theory of superconductivity, the electron-phonon spectral function, which measures the effec-tiveness of phonons of a given energy to scatter electrons onthe Fermi surface, plays a central role. The spectral functionis given by FIG. 4. ͑a͒ Phonon linewidths and ͑b͒ electron-phonon coupling q␯ ͒ប␻
parameters calculated for bcc Li along high-symmetry directions in the Brillouin zone. The coupling parameters ␭q␯ are weighted by a
Within this framework, the mass enhancement parameter is proportional to the inverse-frequency moment of the spectral note longitudinal branches, while open symbols indicate transverse branches. Along the ͓110͔ direction (⌫ to N), the open squares In the linear-response approach, ␦VSCF is computed in the correspond to the T1 branch with polarization along ͓11 process of determining the dynamical matrices. Therefore we open circles correspond to the T2 branch with polarization along calculate the matrix elements on the same grid of phonon ͓001͔. The dashed curve ͑shown only along the ͓100͔ direction͒ wave vectors used in the phonon calculations described in corresponds to linewidths calculated for a free-electron model.
the previous section. The doubly constrained Fermi surfaceaverage of g is computing using Bloch functions at 728 ͑650͒ density of states, the bcc phase has enhanced spectral weight k points in the bcc ͑9R͒ IBZ, with delta functions at the
at low frequencies compared to the close-packed structures.
Fermi level replaced by Gaussians of width 0.02 Ry. All of Since ␭ is proportional to the inverse-frequency moment of our studies of the electron-phonon interaction are carried out F, this difference at low frequencies leads to significant at the experimental volume for bcc Li at 78 K since previous differences in ␭. As can be seen in Table II, we find ␭ work has shown that this yields phonon frequencies that ϭ0.45 for bcc Li, which is in good agreement with earlier agree well with measured dispersion curves.25 Furthermore, first-principles calculations,21–25 and smaller values ranging extrapolations based on available thermal expansion data from 0.33 to 0.37 for the three close-packed structures.
suggest the equilibrium volume changes by less than 0.5% Further evidence for the importance of contributions between Tϭ0 K and Tϭ78 K.37 from low-frequency modes can be found in the values The calculated ␣2F functions are used as input to the of ␻log tabulated in Table II. A logarithmic moment of Eliashberg equations. By solving the Eliashberg equations, ␣2F, this parameter can be expressed as ␻log we obtain the superconducting transition temperature T ϭexp͓͚q␯␭q␯log(␻q␯)/͚q␯␭q␯͔. Although the average pho-
function of the Coulomb pseudopotential ␮*. The Eliash- non frequencies for the different structures differ by only berg equations are solved numerically on the imaginary- about 1%, ␻log is nearly 30% smaller in bcc Li than in the close-packed phases. This indicates that some low-frequency the maximum phonon frequency. The repulsive Coulomb pa- phonon modes in bcc Li couple strongly to the electrons, rameter ␮* is a function of the cutoff frequency, and the thereby dramatically lowering the ␭-weighted logarithmic value used in the numerical solution of the Eliashberg equa- tions is not necessarily what should be used in approximate The wave-vector- and branch-dependent phonon line- widths and coupling parameters for bcc Li are plotted along c equations such as McMillan’s.26 In fact, it is not com- pletely clear which value of ␮* is most appropriate for such high-symmetry directions in Fig. 4͑a͒. The dashed curve along the ͓100͔ direction shows the linewidths calculated for For the results reported here, ␮* is scaled to ␻ϭ␻ a free electron gas with the same average electron density as bcc Li. Within the simple free-electron model, which as- 1/␮*(␻max)ϭ1/␮*(␻c)ϩln(␻c /␻max) sumes a spherical Fermi surface, Thomas Fermi screening, and a single plane wave for the wave function, umklappprocesses are neglected and only longitudinal phonon modes B. Results and discussion
couple to electrons. The model yields a linear q dependence The computed electron-phonon spectral functions for bcc for ␥ at small wave vectors and gives a reasonable descrip- and 9R Li are plotted in Fig. 3͑b͒. The fcc and hcp ␣2F tion of the calculated linewidths for longitudinal modes near curves resemble the 9R results and are omitted for clarity.
the zone center. However, the full calculation finds sizable For each structure, the spectral function is similar in shape to linewidths for transverse modes along all the directions plot- the corresponding phonon density of states, and as with the STRUCTURAL PHASE STABILITY AND ELECTRON- . . .
FIG. 5. ͑a͒ Phonon linewidths and ͑b͒ electron-phonon coupling FIG. 6. Superconducting transition temperature versus ␮* for parameters calculated for bcc Na along high-symmetry directions in bcc and 9R Li. The experimental upper limit on T the Brillouin zone. The coupling parameters ␭ q␯ are weighted by a
modes along ͓␰␰0͔ which contribute so strongly to ␭ in Li.
note longitudinal branches, while open symbols indicate transverse Thus despite similarities in their electronic structure and branches. Along the ͓110͔ direction (⌫ to N), the open squares phonon spectra, bcc Na and bcc Li differ significantly in correspond to the T1 branch with polarization along ͓11 their electron-phonon coupling parameters.
open circles correspond to the T2 branch with polarization along The superconducting transition temperatures calculated ͓001͔. The dashed curve ͑shown only along the ͓100͔ direction͒ for bcc and 9R Li are plotted as functions of ␮*(␻max) in corresponds to linewidths calculated for a free-electron model.
Fig. 6. For bcc Li, the standard value of ␮*ϭ0.12 yieldsT ϭ 0.8 K, similar to earlier results in which estimates for ␭ Figure 4͑b͒ shows that the low-lying T ͓␰␰ were input into the McMillan equation. As can be seen in modes give anomalously large contributions to ␭. Yet Fig.
Fig. 6, a value of ␮*ϭ0.27 is needed in order to suppress Tc 4͑a͒ shows that the linewidths for these modes are compa- below the experimental limit of 6 mK. For 9R Li, the tran- rable in magnitude to those for transverse modes along other sition temperature is found to be consistent with the experi- q␯ is proportional to ␥ q␯ / ␻
mental limit if ␮*ϭ0.19. An approximate upper limit on ␮* does not depend explicitly on ␻q␯ , the large values of ␭q␯
can be obtained by letting ␮ become infinite in the two- square well expression ␮*(␻max)ϭ␮/͓1ϩ␮ ln(EF /␻max)͔.
small phonon frequencies rather than from Fermi surface For all the structures considered here, this condition leads to nesting or other properties related to the electronic structure.
␮*(␻max)Ͻ0.23. This confirms that the large value needed It is interesting to compare these results for bcc Li to to suppress Tc below the experimental limit in bcc Li is most results for bcc Na, which also has a low-lying T ͓␰␰ likely unphysical. For the 9R structure, as well as the other non branch. The phonon dispersion curves of bcc Na are close-packed structures, the electron-phonon interaction is similar to those of bcc Li, with the frequencies scaled by the weak enough to be consistent with the lack of a transition square root of the ratio of atomic masses.39,40 Yet first- above 6 mK, provided that the Coulomb pseudopotential is principles calculations40,41 for bcc Na find ␭Ϸ0.2, which is somewhat larger than is commonly assumed. A realistic cal- less than half the value calculated for bcc Li. Since the de- culation of ␮* is extremely difficult since it requires know- gree of Fermi surface nesting, as measured by ͗͗1͘͘, and the ing the full frequency- and momentum-dependent dielectric electronic density of states, N(EF), are similar in the two response function of the system. In a recent paper, static systems, the differences in electron-phonon coupling arise momentum-dependent dielectric matrices calculated within from differences in the matrix elements. There is a particu- the LDA were used to estimate the Coulomb repulsion pa- larly striking difference in the phonon linewidths along the rameters in bcc and 9R Li.42 The estimates of ␮*Ϸ0.16 for ͓␰␰0͔ direction, as can be seen by comparing Figs. 4 and 5.
both bcc and 9R Li indicate that the standard values assumed For both of the transverse branches along this direction, we for ␮* are not appropriate for this material. This is also q␯M ͔ Li / ͓ ␥ q␯M ͔ Na
supported by a recent study of solutions to the Eliashberg the deeper p pseudopotential in Li arising from the lack of p equation for the electron gas which found that the role of states in the Li core. The stronger potential gives rise to a Coulomb repulsion is significantly underestimated in low- larger change in the bare potential ␦Vbare when atoms are displaced. Furthermore, because the p orbitals are more The thermal effective mass derived from heat capacity tightly bound in Li, the electrons are less effective in screen- measurements provides additional information on the ing ␦Vbare . In combination, these two effects result in sig- electron-phonon mass enhancement parameter. The thermal nificantly larger electron-phonon matrix elements in Li than mass is related to the band mass according to m ϭ in Na. In Na the electrons are able to very effectively screen ϩ␭) and is found experimentally from the electronic contri- the change in ionic potential, particularly for the transverse bution to the specific heat. Measurements for Li yield mth ϭ2.22m, where m is the bare electron mass.20 The band yielding a transition temperature within about 100 K of the mass for 9R Li is calculated to be about 7% larger than that observed starting temperature M s .
for bcc Li, which offsets the difference in ␭. The calculated The thermodynamic properties addressed in this work provide a starting point for understanding the details of the 2.25m and 2.24m for bcc and 9R Li, respectively, are both in good agreement with the specific structural transformation in Li. Many interesting questions heat data. Note however that this analysis does not place remain about the kinetics of the transition, including for ex-ample, details about the reaction path,44 the role played by strict constraints on ␭ since contributions to the mass renor- coherency stresses in determining which close-packed phase malization arising from electron-electron manybody effects appears,4 the appearance of the fcc phase intermediate be- beyond the LDA have been assumed to be negligible.
tween 9R and bcc upon heating,4 and the importance of het- Our results for the electron-phonon interaction in 9R Li are thus consistent both with the observed lack of a super- conducting transition as well as with the specific heat data.
0 ͔ phonon modes which are responsible for One complication is that since Li is never observed to trans- stabilizing the bcc phase at high temperatures also play an form completely to a perfect 9R lattice—there is always a important role in the variation of the electron-phonon cou- mixture of the disordered close-packed polytype, as well as pling strength with crystal structure. It is primarily contribu- some amount of bcc, the exact amounts of which depend on tions from these modes that make the coupling parameter ␭ thermal history—the exact structural characteristics of the about 30% larger in bcc Li than in the close-packed phases.
samples used in the various experiments are not known. De- The coupling in the 9R structure, which is the predominant pending on the details of how grains with different crystal phase at low temperatures, is weak enough to be consistent structures are distributed within the samples, the structural with the current experimental upper limit on Tc , provided disorder itself may play a role in the suppression of super- that a value of ␮*(␻max)Ϸ0.19 is assumed. This is a plau- conductivity in this material. We hope this work motivates sible value given recent theoretical work that shows that the future experimental studies on well-characterized samples of Coulomb repulsion parameter in low-density metals like Li low-temperature Li and on possible connections between the can significantly exceed the standard values assumed.42,43 microstructure of the samples and electronic and transport The calculated enhancement of the electronic mass due to the electron-phonon interaction in 9R Li is also consistent withavailable specific heat data.
IV. CONCLUSIONS
ACKNOWLEDGMENTS
We have used the first-principles linear-response method to compute the vibrational properties of Li in various crystal We thank D.W. Hess and B. Sadigh for helpful discus- structures. The low-energy T ͓␰␰ sions. This work was supported by National Science Foun- correspond to shearing of ͑110͒ planes, distinguish the vibra- dation Grant No. DMR9627778 and by the United States tional properties of the bcc phase from those of the close- Department of Energy, Office of Basic Science, Division of packed phases. In particular, while the close-packed 9R, fcc, Materials Science. A.Y.L. acknowledges the support of the and hcp phases are energetically favorable at Tϭ0, the large Clare Boothe Luce Fund. E.J.N. was supported by the Re- phonon entropy associated with the low-energy modes in bcc search Corporation and by the Natural Sciences and Engi- Li stabilizes the bcc structure upon heating. The quasihar- neering Research Council of Canada. Supercomputing time monic approximation gives a reasonable description of the was provided by the NSF at the Pittsburgh Supercomputing thermodynamics of the transformation at zero pressure, Center and the San Diego Supercomputer Center.
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